Suppose throughout that {pn} is a sequence of non-negative numbers with p0 > 0, that

and that

Let {sn} be a sequence of real numbers.

In this note we shall show that the conditions given by G. Torres in (7) do not suffice to characterize the first Alexander polynomial of a link. We shall recall below Torres' results (for the 2-component case) and state the result of J. H. Bailey on which our argument relies, and then prove the following theorem.

In the following (Q, m, k) will denote a quasi-unmixed local ring of Krull dimension d with maximal ideal m and residue field k. We recall that Q is quasi-unmixed if dim Q̅/p = d for every minimal prime ideal p of the completion of Q.

We improve known Tauberian results concerning the power series method of summability Jp based on the sequence {pn} by removing the condition that pn be asymptotically logarithmico-exponential. We also prove an entirely new Tauberian result for rapidly decreasing pn.

The canonical form of the Riemann tensor is found for certain non-flat empty space-times. Some invariant physical properties of these space-times are investigated. The status of Mach's principle in general relativity theory is briefly discussed in the light of these examples.

A normed algebra A is a pre-B*-algebra if its norm satisfies ∥x*x∥= ∥x∥2 for all elements x ∈ A; if A is also complete in its norm, then A is a B*-algebra (see (l), page 180). In the study of certain locally convex algebras, the problem arose of expressing the condition that an algebra be a pre-B*-algebra in terms of its properties as a locally convex algebra, rather than in terms of the norm. A solution to this problem is presented in this note; the application to the theory of locally convex algebras will appear elsewhere.

In (5), we and Fred Cohen gave some quite general splitting theorems. These described how to decompose the suspension spectra of certain filtered spaces CX as wedges of the suspension spectra of their successive filtration quotients Dq X. The spaces CX were of the form Cr × Xr/(˜) for suitable sequences of spaces {Cr} and {Xr}, and the construction CX was intended to be a reworking in ‘proper generality’ of the constructions introduced in (9).

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:

Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that

(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;

(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

We shall be concerned with the Laguerre polynomials, defined bySome time ago Szegö (8) showed thatfor (a, b, c = 0, 1, …). It is easily deduced from this (cf. (5)) thatfor all A λ ≥ 3.

Considerable advances have been made in recent years in our knowledge of the γ-ray emission of radioactive bodies. It has been established quite definitely that series of definite frequencies are emitted, forming a characteristic line spectrum of the nucleus, and methods of measuring the frequencies have been devised so that at present the γ-ray spectra of most of the radioactive bodies are well known.

Three kinds of proper actions of increasing strength are defined. We prove that the three definitions specialize to the definitions by Bourbaki, by Palais and by Baum, Connes and Higson in their respective settings. The third of these, which thus turns out to be the strongest, originally only concerns actions of second countable locally compact groups on metrizable spaces. In this situation, it is shown to coincide with the other two definitions if the total space locally has the Lindelöf property and the orbit space is regular.

I. Lattice points in the domain | x |α + | y |α ≤ 1

Theorem 4. Let G be the star domain

where α > 0. Then, when α tends to zero,

Proof. The linear substitution

changes G into the similar domain

and so

.

We unconditionally determine $I_{\mathbb Q}(d)$, the set of possible prime degrees of cyclic K-isogenies of elliptic curves with ${\mathbb Q}$-rational j-invariants and without complex multiplication over number fields K of degree ≤ d, for d ≤ 7, and give an upper bound for $I_{\mathbb Q}(d)$ for d > 7. Assuming Serre's uniformity conjecture, we determine $I_{\mathbb Q}(d)$ exactly for all positive integers d.

In this paper we derive the Lorentz-Dirac equation of motion for a charged particle moving in an external electromagnetic field. We use Maxwell's electromagnetic field equations together with the assumptions (1) that all fields are retarded and (2) that the 4-force acting on the charged particle is a Lorentz 4-force. To define the self-field on the world-line of the charge we utilize a contour integral representation for the field due to A. W. Conway. This by-passes the need to define an ‘average field’. In an appendix the case of a scalar field is briefly discussed.

A countably infinite first-order structure M is ω-categorical if every countable N which has the same first-order theory as M is isomorphic to M. By a theorem due independently to Engeler[8], Ryll–Nardzewski[14] and Svenonius[19], this is equivalent to the condition that Aut M has finitely many orbits on Mn for all positive integers n. It is well known that ω-categorical algebraic structures are often intimately related to Boolean algebras. As examples, we mention Theorem 1·3, which is used later, and also the following theorem of J. S. Wilson (proved in [3]), which partly motivates our work.

We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.

We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato–Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ⩽ n11/2(logn)−1/2+o(1) for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n.

In addition, we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato–Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally and might be within reach unconditionally using the currently established potential automorphy.

The literature about the Lorentz transformation is already so wide that some justification is needed for presenting even a small paper on the subject. It is possible to deduce the transformation from essentially different sets of relevant assumptions. The standard text-books on relativity throw little light on this question, which is probably due to the circumstances in which the Lorentz transformation first came into the field. Probably the question is important enough for the following considerations to be worthy of notice.

When the differential analyser (1) is being used it is often necessary to supply to the machine information in the form of a functional relation between variables occurring in the equation. This is usually done by keeping a pointer on a graph on an input table. The x-coordinate lead screw of the input table is driven by the machine, and the y-coordinate lead screw is rotated by hand so as to keep the pointer on the given curve. The disadvantages of this method, namely the necessity for the continuous attention of an operator and the possibility of personal errors, would be obviated by the use of an automatic follower instead of a hand control.

For a bounded open set U ⊂ ℂ, we denote by H∞(U) the collection of all bounded analytic functions on U. We let X denote bdy (U), the boundary of U, Y denote the polynomial hull of U (the complement of the unbounded component of ℂ / X), and U* denote mt (Y), the interior of Y. We denote the sup norm of a function f: A → ℂ by ∥f∥A:

We denote the space of all analytic polynomials by ℂ[z], and we denote the open unit disc by D and the unit circle by S1.